Diversification is measured for portfolios of assets. Portfolios are modeled geometrically and the dimension of the induced geometry is taken as the diversification measurement.
The measurement of total diversification has been absent from the realm of investments and statistics. Diversification is thus left as an inconsistent and qualitatively applied analysis technique and is used implicitly in traditional optimization techniques as a risk mitigating control.
Diversification is a powerful tool that reduces the variance of a portfolio and consequently helps to stabilize performance, potentially enabling more consistent returns and mitigating risk. Diversification has also been shown to be a cornerstone of judging the prudence of a fiduciary.
Diversification is an investment attribute that generates great consensus as to its efficacy as a management attribute. Therefore, the utility of a robust and consistent measurement of diversification would be strong. Despite the widespread acceptance of diversification, investors suffer from the absence of a well-defined, uniform quantitative metric. In fact, investors are accustomed to thinking of diversification as only an abstract or qualitative attribute.
A statistical measurement of the relationship of assets is an indication of diversification. Yet, these relationships, traditionally co-variances or correlations, measure a unique relationship between any two single assets in the portfolio. Because the portfolio represents the entire composition of all these relationships, the measure of any one single relationship fails to represent a genuine level of portfolio diversification.
Another common misinterpretation of diversification is Beta. Beta is a measurement given to a portfolio that describes the amount of performance of that portfolio that can be explained by the market forces. While Beta can be construed as a measurement of diversification, it too has some limitations. Beta requires the presence of another portfolio external to that portfolio being measured. Conventionally, this portfolio may be the S&P 500 or other broad index. Beta thus has some utility for measuring the degree to which an asset will move with the market. This index is thus an approximation of the market.
Beta has several problems, in that defining the market is inherently problematic. In the United States alone, there are more than 200,000 investment products. No investment index can even come close to accounting for all of the different investment possibilities. Ultimately, every new IPO, every new start-up business and every new idea defines that market. No relative measure can ever capture the true market.
A further issue is that there is little efficacy in comparing the portfolio to the market. Beta is a relative measurement, whereas a measurement of diversification that would better help investors to construct and manage portfolios for performance purposes would not be concerned with items external to the portfolio but only those assets comprising the portfolio. Therefore, a holistic measurement of diversification that was independent of any market benchmark or index would be desirable by investors seeking better performance.
Investors care more about what happens to their portfolio than what the market does. Investors who have a sole focus on absolute returns are endeavoring to maximize the value of their portfolio. Such investors would prefer a holistic measurement of diversification than a measure that is relative to another (and essentially arbitrary) index.
Diversification has been primarily measured as to the number of assets held and to a lesser extent the largest allocations among those assets. Measuring diversification in this way fails to account for disparate weightings of assets and fails to account for the commonality of assets. To illustrate, consider a 10-asset portfolio consisting of ten equally weighted portfolios, having each asset perform identically to another asset. This portfolio has the overall performance of only one asset, despite holding ten different investments.
Traditional and commonly used statistics can provide some insight into diversification, but they fail to measure it. For example, measures of dispersion, central tendency and distribution are in one sense measures of diversification. Perhaps the most applicable among these measurements is the kurtosis. The kurtosis is a value reflecting how close observed values are to the mean of those values. The kurtosis is the fourth moment of a probability distribution and is therefore a dimensionless quantity. This measure also fails as a robust measurement of diversification in that it cannot account for the full dimensionality of the underlying data. In the conventional application, a portfolio manager would use the distribution and kurtosis to view the portfolio over time, and thus it does not provide any holistic insight.
Cluster analysis has some useful applications to help analyze diversification. However, cluster analysis fails to distill portfolio diversification to a singular value that may then be used to aid in the relative analysis of portfolios. Distilling portfolio diversification to one singular value is thus desirable to aid in investment analysis and selection, optimization, attribution and presentation of portfolios and assets.
Investors currently have no way of measuring diversification except for the Concentration co-Efficient (CC), Intra-Portfolio Correlation (IPC), and their derivates. A consistent, robust and quantitative diversification metric is thus of great utility to the industry.
Concentration Coefficient (CC)
The concentration coefficient (CC) measures portfolio concentration in terms of the asset weightings. In an equal weighted portfolio, the CC will be equal to the number of assets. As the portfolio becomes more concentrated in particular assets the CC will be proportionally reduced.
Thus, the Concentration Coefficient (CC) is defined as:
      CC    t    P    ≡            (                        ∑                      i            =            1                    N                ⁢                              (                          w                              i                ,                t                            P                        )                    2                    )              -      1      
P is the portfolio
N is the number of stocks held in the portfolio
Wi,t is the weight of the ith stock in the portfolio at time t
The concentration coefficient has the desirable property of being a discrete measurable quantity. However, it fails to account for the relationships of assets and thus is inadequate for managing diversification against market risk or systemic risk, which are among the most prevailing risks investors face, and need to be managed.
Intra-Portfolio Correlation (IPC)
Intra-portfolio correlation (IPC) is a means to quantify diversification. The range is from −1 to 1, with values approaching 1 being the least diversified. The IPC is a weighted average intra-portfolio correlation.
The Intra-Portfolio Correlation (IPC) statistic is calculated as follows:
  IPC  =            ∑      i        ⁢                  ∑        j            ⁢                        X          i                ⁢                  X          j                ⁢                  p          ij                                    Xi is the fraction invested in asset i        Xj is the fraction invested in asset j        Pij is the correlation between assets i and j        The expression may be computed when i≠j        
The IPC is thus a measure of diversification against risks such as systemic risk but fails to account for other risks such as security risk, concentration risk and model risk.
Other Forms
Other forms of diversification analysis pertain to the classification of an asset such as depicting the sector or asset class assigned to portfolio assets. Classification schemas may be economic sectors, industries, valuation models or geographic locations. In addition, and especially within corporate portfolio models, elements may be product lines, the difference between these product lines in terms of elements such as price, characteristics of the targeted market, manufacturing style, goods or services, and materials used. Corporations also seek diversification for their investor base, supplier base, employee base, customer base. All such analysis techniques are qualitative, not quantitative.
While useful, this analysis fails to entirely account for diversification by subjugating analysis to only the studied elements, and again failing to provide a single measurement.
These techniques have additional drawbacks. They require categorization of all components. Frequently, the categorization of such assets lacks rigor and suffers from a non-optimal division. Additionally, there is no consistent process or categorization for determining asset classes, which portends a lack of consistency when describing the diversification attributes of various portfolios. Therefore characterizing diversification as having exposure to various asset classes results in inconsistent, non-comparable, and varying solutions.
A further element of subjectivity often results when mapping investments to categories. The inconsistent mapping process of various institutions map results in dissimilar diversification analyses.
These techniques fail to deliver one simple numerical value that an investor can use to analyze and compare portfolios. Therefore the result of the analysis is inadequate to compare statistically similar portfolios as the investor would have to analyze an array of values and most humans are incapable of accurately determining an optimal or even superior result, wherein if the result is reduced to one value, its comprehension and comparative efficacy increase dramatically.
Types of Diversification
For business strategy purposes, diversification is sometimes categorized as vertical, horizontal or concentric. In these conventional meanings, horizontal diversification is meant to mean broadening of the product line. Vertical diversification is the integration of the supply chain or distribution outlets and concentric diversification is a corporate growth strategy whereby a business builds its total sales by acquiring or establishing other unrelated businesses that may share management or technical efficiencies.
When the decision maker has a presumption of control, a diversification strategy can have several disadvantages. Namely as a corporation diversifies, fewer resources are able to be devoted to the same assets. This can diminish the ability for any one asset to reach a critical mass. A more microeconomic view may be that an asset with certain fixed and sunk costs may have a decreased or negative net present value when other assets supporting the continued development and sales of that asset are diluted by the new diversified strategy. One could imagine that certain efficiencies and advantages can be obtained by practicing the discipline of focus.
While the efficacy of maximizing diversification is not clear in corporate models, the utility of measuring diversification is clear. Accounting for diversification germane to a company provides the company with valuable information that when combined with business intelligence can shape strategy and execution. Product lines, target customers and business lines may be better optimized to increase diversification and decrease macroeconomic risks, or deliberately focused to exploit expected advantages. Shareholders of companies also stand to benefit from an internal diversification measurement; such values provide insight as to the focus, interrelationships and vulnerabilities of the company.
The only limitation towards the efficacy of investment portfolio diversification stems from a belief in the relative attractiveness of a particular asset. If other assets of comparable attractiveness cannot be discovered, then there is a reasonable justification for holding less diverse portfolios. This constraint is really rather a constraint upon the fund manager and the fund manger's resources including her own time for studying, analyzing and predicting returns on investments. As an investment manager's resources grow, the manager's ability to discover several highly attractive assets also increases and thus within a framework of rational participants and increasingly efficient markets the greater the importance of diversification in the investment policy and investment process. Without the ability to intelligently predict basic portfolio optimization input assumptions, any investors would be foolish to deliberately accept a portfolio of less diversification.